Finding a Matrix's Inverse with Elementary Matrices

# Finding a Matrix's Inverse with Elementary Matrices

Recall that an elementary matrix $E$ performs an a single row operation on a matrix $A$ when multiplied together as a product $EA$. If $A$ is an $n \times n$ matrix, then we can say that $A$ is constructed from applying a finite set of elementary row operations on $I_n$. We first take a finite set of elementary matrices $E_1, E_2, ..., E_k$ used to reduce $A$ to $I$:

(1)
$$E_{k}E_{k-1}...E_2E_1A = I_n$$

If we take this equation and multiply the $E_i$'s by their inverses $E_i^{-1}$ successively on the left, we get that:

(2)
$$A = E_{1}^{-1}E_{2}^{-1}...E_{k}^{-1}I_n$$

If we take the earlier formula and multiply the equation on the right by $A^{-1}$, it also follows that:

(3)
$$A^{-1} = E_{k}E_{k-1}...E_2E_1A$$

We can apply these formulas to help us find $A$ or $A^{-1}$ whenever we need it.

# Using Elementary Matrices to Invert a Matrix

Suppose that we have an invertible matrix $A$. If we append $A$ to $I$ in the manner: $[A \mid I ]$ and reduce $A$ to $I$, we will effectively produce $I$ to $A^{-1}$, that is:

(4)
\begin{align} [A \mid I] \to [I \mid A^{-1}] \end{align}

For example, consider the following matrix:

(5)
\begin{align} A = \begin{bmatrix} 2 & -1 & 4\\ 1 & 1 & 1\\ 3 & -2 & 1 \end{bmatrix} \end{align}

First let's append the identity matrix to $A$ such that we obtain $[A \mid I ]$:

(6)
\begin{bmatrix} 2 & -1 & 4 & 1 & 0 & 0\\ 1 & 1 & 1& 0 & 1 & 0\\ 3 & -2 & 1 & 0 & 0 & 1 \end{bmatrix}

We will proceed with the following row operations to reduce $A$ to $I$ (omitting details), which will result in turning $I$ to $A^{-1}$:

(7)
\begin{bmatrix} 1 & 0 & 0 & -\frac{3}{16} & \frac{7}{16} & \frac{5}{16} \\ 0 & 1 & 0 & -\frac{1}{8} & \frac{5}{8} & -\frac{1}{8}\\ 0 & 0 & 1 & \frac{5}{16} & -\frac{1}{16} & -\frac{3}{16} \end{bmatrix}

From this reduction we obtain that $A^{-1} = \begin{bmatrix} -\frac{3}{16} & \frac{7}{16} & \frac{5}{16} \\ -\frac{1}{8} & \frac{5}{8} & -\frac{1}{8}\\ \frac{5}{16} & -\frac{1}{16} & -\frac{3}{16} \end{bmatrix}$.